openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Models /GroundMainFinalAssignmentLabelInjections.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundMainFinalPersistentAssignmentLabels | |
| namespace OAI | |
| namespace PartitionPilot.Forcing.GroundMainPrefix.FinalSupportFulfillment | |
| open SetFormula SetRealization | |
| universe u v | |
| variable {M : TransitiveGround.{u}} {K Labels C D rD : ZFSet.{u}} | |
| variable {pairing : Realize M.sets SetFormula.pairing Fin.elim0} | |
| variable {unions : Realize M.sets SetFormula.union Fin.elim0} | |
| variable {powers : Realize M.sets powerSet Fin.elim0} {separation : SeparationSchema M.sets} | |
| variable {I : Type v} [Small.{u} I] [Preorder I] | |
| variable {run : GroundMainPrefix M K Labels C D rD pairing unions powers separation I} | |
| variable {level : GroundMainCandidateLevel M K} {i₀ : I} | |
| variable (fulfilled : run.FinalSupportFulfillment level i₀) | |
| theorem structural_successor_assignment_label_injection | |
| (collection : CollectionSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (labels : Labels = M.groundOmegaOne pairing unions powers separation collection infinite) | |
| (small : M.sets.Countable) (smallD : M.groundEmbeddable D ZFSet.omega) | |
| (irreflexive : ∀ q ∈ D, ZFSet.pair q q ∉ rD) | |
| (columns : ZFSet.{u}) (hColumns : columns ∈ M.sets) | |
| {mu : ZFSet.{u}} (hmu : mu ∈ M.sets) (muOrdinal : ordinalSetProperty mu) : | |
| let geometry := run.finalGeometry i₀ | |
| letI := geometry.group_code.toGroup | |
| letI := geometry.meet.order.toPartialOrder | |
| letI := geometry.decodedBottom | |
| let s := SetMembers.presentation geometry.carrier | |
| let b := SetMembers.presentation geometry.actors | |
| let a := geometry.decodedAction | |
| let directed := geometry.decoded_directed fulfilled.directed | |
| let hL := geometry.carrier_ground | |
| let hG := geometry.actors_ground | |
| let hC := hColumns | |
| let order : s.conditionOrder ∈ M.sets := geometry.decoded_order_eq.symm ▸ geometry.order_ground | |
| let graph : a.actionGraph s b ∈ M.sets := geometry.decoded_action_eq.symm ▸ geometry.action_ground | |
| let nodeMaps := M.conditionMapFamily_mem_of_actionGraph a s b pairing unions powers separation | |
| collection hL hG graph | |
| let supports := geometry.decoded_supportBasis_ground pairing unions powers separation | |
| letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| letI := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| let act := M.internalRowAction a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier columns ⊤ | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode ⊤) mu s.nonbottom.carrier rows.sUnion.sUnion rows S | |
| ∀ valid : ∀ z ∈ T, IsName c z, | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let squared := orderedPairNameFamily (c.pairEmpty.encode ⊤) lifted lifted | |
| ∀ (hLifted : lifted ∈ M.sets) (hSquared : squared ∈ M.sets) | |
| (valid1 : ∀ z ∈ lifted, IsName c.pairEmpty z) | |
| (valid2 : ∀ z ∈ squared, IsName c.pairEmpty z) | |
| (closed1 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ lifted → (act.act c.pairEmpty g tau).val ∈ lifted) | |
| (closed2 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ squared → (act.act c.pairEmpty g tau).val ∈ squared) | |
| (children1 : ∀ tau : Name c.pairEmpty, tau.val ∈ lifted → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (children2 : ∀ tau : Name c.pairEmpty, tau.val ∈ squared → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (stages history : ZFSet.{u}) (hs : stages ∈ M.sets) (hh : history ∈ M.sets) | |
| (stagesOrdinal : ordinalSetProperty stages) | |
| (trace : OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder (c.encode ⊤) (act.actionGraph c b) | |
| b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared level prior) stages history), | |
| let coordinates := M.prod_mem pairing unions powers separation (M.nonbottom_carrier_mem s separation hL) hC | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite coordinates two | |
| let forcingOrder := M.countableFunctionOrder_mem pairing unions powers separation infinite coordinates two | |
| let forcingAction := M.internalRowActionGraph_mem a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let omega := M.omega_mem separation infinite | |
| let topMem := c.encode_mem ⊤ | |
| let topRows : ∀ r ∈ c.carrier, ZFSet.pair r (c.encode ⊤) ∈ c.conditionOrder := | |
| fun r hr => by | |
| obtain ⟨r, rfl⟩ := (c.mem_carrier r).1 hr | |
| exact (c.pair_mem_conditionOrder r ⊤).2 le_top | |
| let packed := M.structuralRefinementHistory b.carrier s.carrier s.conditionOrder | |
| (supportBasisCode s b a.principalFix) lifted squared stages history | |
| ∀ (alpha : ZFSet.{u}) (member : alpha ∈ stages), | |
| let hAlpha := M.transitive hs member | |
| let cutOrdinal := ordinalSetProperty_member stagesOrdinal member | |
| let subset := (And.left stagesOrdinal) alpha member | |
| let lowerPacked := graphRestriction alpha packed | |
| let Q := M.refinementHistoryForcing c.carrier lifted alpha lowerPacked | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code lifted alpha lowerPacked | |
| let historyRun := M.structuralForcingTrace_recursiveOrder c.carrier c.conditionOrder (c.encode ⊤) | |
| (act.actionGraph c b) b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared stagesOrdinal trace | |
| letI := M.refinementHistoryOrderTop c.conditionOrder_code lifted alpha lowerPacked | |
| (historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier (c.encode ⊤) topMem topRows | |
| let lower := SetMembers.presentation Q | |
| let localAct := M.semanticStructuralPrefixAction c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hh trace hAlpha cutOrdinal subset | |
| let localValid := M.refinementHistoryRowNameFamily_valid c | |
| (historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier lifted valid1 | |
| let upperSet := M.refinementHistoryForcing c.carrier lifted stages packed | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed | |
| letI := M.refinementHistoryOrderTop c.conditionOrder_code lifted stages packed | |
| historyRun.carriers stagesOrdinal carrier (c.encode ⊤) topMem topRows | |
| let upper := SetMembers.presentation upperSet | |
| let hPacked := M.structuralRefinementHistory_mem pairing unions powers separation collection | |
| ![b.carrier, s.carrier, s.conditionOrder, supportBasisCode s b a.principalFix, | |
| lifted, squared, stages, history] | |
| (Fin.cases hG (Fin.cases hL (Fin.cases order (Fin.cases supports | |
| (Fin.cases hLifted (Fin.cases hSquared (Fin.cases hs (Fin.cases hh (fun j => Fin.elim0 j))))))))) | |
| let witnesses := M.structuralForcingTrace_witnesses c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace | |
| let mono := M.structuralHistory_rawTests_mono b s a ⊥ directed lifted squared stages history | |
| pairing unions powers separation collection hG hL supports hh hLifted witnesses | |
| let inc := RecursiveRefinementOrderHistory.inclusionFunction historyRun stagesOrdinal cutOrdinal subset | |
| pairing unions powers separation infinite choice hs hPacked hLifted hAlpha carrier (c.encode ⊤) | |
| topMem topRows mono | |
| ∀ (_successor : insert alpha alpha = stages) | |
| (V E : SetMembers geometry.carrier) (U Rf Rc : (M.nameDomain lower).names), | |
| let idx := refinementIndexCode alpha (s.encode V) (s.encode E) U.val.val Rf.val.val Rc.val.val | |
| idx ∈ M.niceRefinementIndices localAct lower s a alpha lifted squared localValid → | |
| ∀ (H : ConditionFilter (SetMembers upperSet)), H.GenericFor (formulaFamily (M.nameDomain upper)) → | |
| let label := Name.refinementAssignmentLabel upper inc ⊤ lifted localValid idx Rf.val Rc.val | |
| let J := Name.eval {p | H.mem p} label | |
| let coarse := Name.eval {q | H.mem (inc q)} (Name.refinementQuotient lifted localValid U.val Rc.val) | |
| let fine := Name.eval {q | H.mem (inc q)} (Name.refinementQuotient lifted localValid U.val Rf.val) | |
| let upperAct := M.semanticStructuralAction c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace | |
| InjectionCode coarse fine J ∧ | |
| J ∈ (M.symmetricExtension upper upperAct (a.principalSupportFamily ⊥ directed) H).sets := by | |
| let geometry := run.finalGeometry i₀ | |
| let := geometry.group_code.toGroup | |
| let := geometry.meet.order.toPartialOrder | |
| let := geometry.decodedBottom | |
| let s := SetMembers.presentation geometry.carrier | |
| let b := SetMembers.presentation geometry.actors | |
| let a := geometry.decodedAction | |
| let directed := geometry.decoded_directed fulfilled.directed | |
| let hL := geometry.carrier_ground | |
| let hG := geometry.actors_ground | |
| let hC := hColumns | |
| let order : s.conditionOrder ∈ M.sets := geometry.decoded_order_eq.symm ▸ geometry.order_ground | |
| let graph : a.actionGraph s b ∈ M.sets := geometry.decoded_action_eq.symm ▸ geometry.action_ground | |
| let nodeMaps := M.conditionMapFamily_mem_of_actionGraph a s b pairing unions powers separation | |
| collection hL hG graph | |
| let supports := geometry.decoded_supportBasis_ground pairing unions powers separation | |
| let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| let := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| let act := M.internalRowAction a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier columns ⊤ | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode ⊤) mu s.nonbottom.carrier rows.sUnion.sUnion rows S | |
| dsimp only | |
| intro valid | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let squared := orderedPairNameFamily (c.pairEmpty.encode ⊤) lifted lifted | |
| intro hLifted hSquared valid1 valid2 closed1 closed2 children1 children2 | |
| stages history hs hh stagesOrdinal trace | |
| let coordinates := M.prod_mem pairing unions powers separation (M.nonbottom_carrier_mem s separation hL) hC | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite coordinates two | |
| let forcingOrder := M.countableFunctionOrder_mem pairing unions powers separation infinite coordinates two | |
| let forcingAction := M.internalRowActionGraph_mem a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let omega := M.omega_mem separation infinite | |
| let topMem := c.encode_mem ⊤ | |
| let topRows : ∀ r ∈ c.carrier, ZFSet.pair r (c.encode ⊤) ∈ c.conditionOrder := | |
| fun r hr => by | |
| obtain ⟨r, rfl⟩ := (c.mem_carrier r).1 hr | |
| exact (c.pair_mem_conditionOrder r ⊤).2 le_top | |
| let packed := M.structuralRefinementHistory b.carrier s.carrier s.conditionOrder | |
| (supportBasisCode s b a.principalFix) lifted squared stages history | |
| intro alpha member | |
| let hAlpha := M.transitive hs member | |
| let cutOrdinal := ordinalSetProperty_member stagesOrdinal member | |
| let subset := (And.left stagesOrdinal) alpha member | |
| let lowerPacked := graphRestriction alpha packed | |
| let Q := M.refinementHistoryForcing c.carrier lifted alpha lowerPacked | |
| let := M.refinementHistoryPartialOrder c.conditionOrder_code lifted alpha lowerPacked | |
| let historyRun := M.structuralForcingTrace_recursiveOrder c.carrier c.conditionOrder (c.encode ⊤) | |
| (act.actionGraph c b) b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared stagesOrdinal trace | |
| let := M.refinementHistoryOrderTop c.conditionOrder_code lifted alpha lowerPacked | |
| (historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier (c.encode ⊤) topMem topRows | |
| let lower := SetMembers.presentation Q | |
| let localAct := M.semanticStructuralPrefixAction c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hh trace hAlpha cutOrdinal subset | |
| let localValid := M.refinementHistoryRowNameFamily_valid c | |
| (historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier lifted valid1 | |
| let upperSet := M.refinementHistoryForcing c.carrier lifted stages packed | |
| let := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed | |
| let := M.refinementHistoryOrderTop c.conditionOrder_code lifted stages packed | |
| historyRun.carriers stagesOrdinal carrier (c.encode ⊤) topMem topRows | |
| let upper := SetMembers.presentation upperSet | |
| let hPacked := M.structuralRefinementHistory_mem pairing unions powers separation collection | |
| ![b.carrier, s.carrier, s.conditionOrder, supportBasisCode s b a.principalFix, | |
| lifted, squared, stages, history] | |
| (Fin.cases hG (Fin.cases hL (Fin.cases order (Fin.cases supports | |
| (Fin.cases hLifted (Fin.cases hSquared (Fin.cases hs (Fin.cases hh (fun j => Fin.elim0 j))))))))) | |
| let witnesses := M.structuralForcingTrace_witnesses c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace | |
| let mono := M.structuralHistory_rawTests_mono b s a ⊥ directed lifted squared stages history | |
| pairing unions powers separation collection hG hL supports hh hLifted witnesses | |
| let inc := RecursiveRefinementOrderHistory.inclusionFunction historyRun stagesOrdinal cutOrdinal subset | |
| pairing unions powers separation infinite choice hs hPacked hLifted hAlpha carrier (c.encode ⊤) | |
| topMem topRows mono | |
| intro successor V E U Rf Rc index H ordinary | |
| have total := fulfilled.structural_successor_assignment_label_total collection infinite choice labels | |
| small smallD irreflexive columns hColumns hmu muOrdinal valid hLifted hSquared | |
| valid1 valid2 closed1 closed2 children1 children2 | |
| stages history hs hh stagesOrdinal trace alpha member successor V E U Rf Rc index H ordinary | |
| exact M.semanticStructuralAssignmentLabel_injection_package c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace member infinite choice V E U Rf Rc index H ordinary total | |
| end PartitionPilot.Forcing.GroundMainPrefix.FinalSupportFulfillment | |
| end OAI | |